Definition:Composition Series

Definition

Let $G$ be a finite group.

Definition 1

A composition series for $G$ is a normal series for $G$ which has no proper refinement.

Definition 2

A composition series for $G$ is a sequence of normal subgroups of $G$:

$\set e = G_0 \lhd G_1 \lhd \cdots \lhd G_n = G$

where:

$G_{i - 1} \lhd G_i$ denotes that $G_{i - 1}$ is a proper normal subgroup of $G_i$

such that:

for all $i \in \set {1, 2, \ldots, n}$, $G_{i - 1}$ is a proper maximal normal subgroup of $G_i$.

Composition Length

Let $\HH$ be a composition series for $G$.

The composition length of $G$ is the length of $\HH$.

Composition Factor

Let $\set e = G_0 \lhd G_1 \lhd \cdots \lhd G_{n - 1} \lhd G_n = G$ be a composition series for $G$.

Each of the quotient groups:

$G_1 / G_0, G_2 / G_1, \ldots, G_n / G_{n - 1}$

are the composition factors of $G$.

Examples

Cyclic Group $C_8$

There is $1$ composition series of the cyclic group $C_8$, up to isomorphism:

$\set e \lhd C_2 \lhd C_4 \lhd C_8$

Quaternion Group $Q$

There are $2$ composition series of the quaternion group $Q$, up to isomorphism:

$\set e \lhd C_2 \lhd C_4 \lhd Q$

$\set e \lhd C_2 \lhd K_4 \lhd Q$

where:

$C_n$ denotes the cyclic group of order $n$.

$K_4$ denotes the Kline $4$-group.

Dihedral Group $D_4$

There are $2$ composition series of the dihedral group $D_4$, up to isomorphism:

$\set e \lhd C_2 \lhd C_4 \lhd D_4$

$\set e \lhd C_2 \lhd K_4 \lhd D_4$

where:

$C_n$ denotes the cyclic group of order $n$.

$K_4$ denotes the Kline $4$-group.

Dihedral Group $D_6$

There are $3$ composition series of the dihedral group $D_6$, up to isomorphism:

$\set e \lhd C_3 \lhd C_6 \lhd D_6$

$\set e \lhd C_2 \lhd C_6 \lhd D_6$

$\set e \lhd C_3 \lhd D_3 \lhd D_6$

where $C_n$ denotes the cyclic group of order $n$.

Also see

• Equivalence of Definitions of Composition Series

• Composition Series of Group of Prime Power Order

• Results about composition series can be found here.

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Conjugacy, Normal Subgroups, and Quotient Groups: $\S 52$